On the spectrum of invertible semi-hyponormal operators (Q2476170)

Let \(T\) be a semi-hyponormal operator with polar decomposition \(T=U| T| \), and let \((T_k)_{0\leq k\leq 1}\) be the family of the general polar symbols of \(T\) defined by \[ T_k:=kT_++(1-k)T_-, \quad 0\leq k\leq 1, \] \(T_+:=\text{st}-\lim_{n\to+\infty}{U^*}^nTU^n\) and \(T_-:=\text{st}-\lim_{n\to+\infty}U^nT{U^*}^n\). In [J.~Oper.\ Theory 5, 257--266 (1981; Zbl 0477.47019)], \textit{D.--X.\thinspace Xia} proved, using the singular integral model, that the spectrum \(\sigma(T)\) of \(T\) coincides with the union of the spectra of the general polar symbols of \(T\), i.e., \(\sigma(T)=\bigcup_{0\leq k\leq1}\sigma(T_k)\) [see also ``Spectral theory of hyponormal operators'' (Oper.\ Theory, Adv.\ Appl.\ 10, Birkhäuser, Basel-Boston-Stuttgart) (1983; Zbl 0523.47012); Sci.\ Sin.\ 23, 700--713 (1980; Zbl 0441.47036)]. In the paper under review, the author provides a different proof of the same result for the case when \(T\) is an invertible semi-hyponormal operator without using the singular integral model.